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Single-molecule biomechanical measurements, such as the force to unfold a protein domain …
where nij is the number of i j transitionsdirectly calculated from the adhesion score sequence; p10 +p11 = 1 and p00 + p01 = 1. By definition, p01 is the probabilityof having adhesion (i.e., Xi+1 = 1) under the condition thatthe previous adhesion test is not successful (i.e., Xi = 0).It is also given a short notation p by definition. p11 is theprobability of having adhesion in the next test if the previoustest also results in adhesion. It is also given a short notationp + p by definition. p represents an increment (positive ornegative) in the probability of adhesion in the (i + 1)th testdue to the occurrence of adhesion in the ith test, which canbe thought of as a memory index. If the i.i.d. assumption holds,p01 = p11 = p and p = 0, and we recover the Bernoulli sequence.p can be estimated from the average adhesion frequency becauseit also equals Pa. If the i.i.d. assumption is violated (p01p11 and p 0), p will not be equal to Pa (see Eq. 5 below).
Direct calculations of p01 and p11 for the data in Fig. 2 D–Fshow that their values are close to each other and to the averagedadhesion probability, Pa, for LFA-1/ICAM-1 interaction, providingpreliminary validation for the i.i.d. assumption. By comparison,transition 1 1 is more favorable than transition 0 1 for theTCR/pMHC interaction, whereas the opposite seems true for theC-cadherin interaction, indicating that the i.i.d. assumptionmight be violated for these two cases.
Closer inspection of the scaled adhesion scores in Fig. 2 D–Freveals that they are clustered at different sizes. It seemsintuitive that, for a given "cluster size" m (i.e., m consecutiveadhesions), the number of times it appears in an adhesion scoresequence contains statistical information about that sequence.Our intuition is supported by the data in Fig. 3, which showsthe cluster size distribution enumerated from the adhesion scoresequences in Fig. 2. Compared with the distribution for theLFA-1/ICAM-1 interaction (Fig. 3A), the distribution for theTCR/pMHC interaction (Fig. 3B) has more clusters of large size,whereas the distribution for the C-cadherin interaction (Fig. 3C)has more single adhesion events surrounded by no-adhesion events.
To quantify the differences among the three cases in Fig. 3,we derived a formula to express the number, MB, of clustersof size m expected in a Bernoulli sequence of length n and probabilityp for the positive outcome in each test:
The first term in the upper branch on the right-handside of Eq. 2 represents summation over the probabilities ofhaving a cluster of size m in all possible positions, i.e.,for clusters starting at i = 2 to i = n – (m – 1).Clusters starting from X1 or ending with Xn are accounted forby the second term in the upper branch. The lower branch ofour formula accounts for the sequence of all 1s. It becomesapparent from the above derivation that Eq. 2 assumes equalprobability for the cluster to take any position in the sequence.This can be thought of as a stationary assumption.
The total number of positive adhesion scores in the entire sequencecan be calculated by multiplying Eq. 2 by m and then summingover m from 1 to n. It can be shown by direct calculation thatmMB(m, n, p) = np. Here, np is the expected total number ofadhesion events. This outcome is predicted and shows that Eq. 2is self-consistent.
MB is plotted vs. Pa (= p) in Fig. 4A for n = 50 and for clustersof sizes m = 1–4. The actual cluster size distributionsenumerated from measured adhesion score sequences (e.g., Fig. 3A)should be realizations of the underlying stochastic process.We used computer simulations to characterize the statisticalproperties of this stochastic process. The mean ± SEMof the number of clusters of size 1 from 3 (open triangles,mimic experiments where three to five pairs of cells were usuallytested) and 50 (filled triangles, a good approximation to Eq. 2)simulated Bernoulli sequences for several Pa values are shownin Fig. 4A, which evidently agree well with MB(1, 50, Pa) givenby Eq. 2.
We next extend Eq. 2 to the case of a Markov sequence by includinga single-step memory. The four conditional probabilities definedin Eq. 1 form a one-step transition probability matrix [P] ofa stationary Markov sequence. Using Bayes' theorem for totalprobability, the unconditional probabilities for the (i + 1)thtest are related to those for the ith test by [P]:
By applying the Chapman–Kolmogorov equation(8), the n-step transition matrix can be obtained as [P(n)]= [P]n. With the initial condition of P(X1 = 1) = p and P(X1= 0) = 1 – p, it can be shown by mathematical inductionthat P(Xi = 1) = p(1 – pi)/(1 – p). The formulafor the expected cluster size distribution can now be extendedas follows:
Setting p = 0 reduces Eq. 4 to Eq. 2, as required. Eq. 4 hasanother special case at p = 1 – p when it simplifies toMM(m, n, p, 1 – p) = p(1 – p)n–m, which describesthe situation where the first adhesion event in the sequencewould increase the probability for adhesion in the next testto 1, leading to continuous adhesion for all tests until theend of the experiment.
If p = 0 (i.e., a Bernoulli sequence), Fn approaches p as nbecomes large. However, if p 0 (i.e., a Markov sequence), thenFn approaches p/(1 – p).
MM is plotted vs. Pa (related to p and p by Eq. 5) in Fig. 4Bfor n = 50, m = 1 (i.e., solitary 1s bound by 0s from both ends)for p ranging from –0.3 to 0.5 in increments of 0.1. Thecase of p = 0 (i.e., Bernoulli sequence) is plotted as a solidcurve. Cases of p > 0 (i.e., memory with positive feedback)are plotted as dotted curves. Cases of p < 0 (i.e., memorywith negative feedback) are plotted as dashed curves. Increasingp shifts the curve downward toward a smaller number of isolated1s in an adhesion score sequence.
It can be shown by direct calculation that mMM(m, n, p, p) = p[n – p(1 – pn)/(1– p)]/(1 – p). Here, the left-hand side sums adhesionevents distributed in various clusters of different sizes, andthe right-hand side is the expected number of adhesion eventsin n repeated tests, nE(Fn) (cf. Eq. 5). This predicted resultconfirms that Eq. 4 is self-consistent.
In Fig. 3, Eq. 4 was fit (curves) to the measured cluster sizedistributions (bars) (see Materials and Methods for proceduredetail). Three different types of behaviors can be clearly discerned.A memory index p 0 was returned from fitting the LFA-1/ICAM-1data in Fig. 3A. Because of the limited amount of data, smallfluctuations of p from zero could be observed even for Bernoullisequences, as seen from computer simulations of a small numberof cell pairs. Fitting the TCR/pMHC data (Fig. 3B) returneda positive p, indicating the presence of memory with positivefeedback, whereas fitting the C-cadherin data (Fig. 3C) returneda negative p, indicating the presence of memory with negativefeedback. These results support our preliminary conclusionsbased on the observations in Fig. 2 and preliminary analysisusing the p01, p11, and Pa comparison.
The above fittings used the distribution of clusters of allsizes (i.e., all m values) for a given Pa to evaluate p. However,differences in the distribution of cluster sizes are dominatedby the difference in the expected number of clusters of size1 (Fig. 3, comparing the first bar in each panel). Thus, thenumber of clusters of size 1 in experimental adhesion scoresequences can be used as a simple indicator to evaluate thememory effect.
Analysis so far has used the raw data shown in Fig. 2, whichhave similar Pa values. To obtain further support with 10x moredata, we varied contact times and/or receptor–ligand densitiesto obtain different average adhesion frequencies, which alsoallowed us to examine the potential effects of molecular densitieson p through Pa. For each system, the experimental number ofclusters of size 1, Mexp(1), for each Pa value was plotted inFig. 4B to compare with the theoretical curves, MM[1, 50, p(50,Pa, p), p]. It can be seen that the LFA-1/ICAM-1 data are scatteredevenly from both sides of the solid curve corresponding to p= 0. By comparison, almost all of the TCR/pMHC data are belowthe theoretical curve for p = 0, and most of the C-cadherindata are above that curve. These results further support ourconclusions regarding three types of behaviors.
The memory index p was obtained from fitting experimental clustersize distribution with Eq. 4 and is plotted in Fig. 5(solidbars) along with p estimated from direct calculation (usingEq. 6 below) (open bars). Comparable results were obtained byboth approaches for all Pa values tested for all three systemsexhibiting qualitatively distinct behaviors. p values for theLFA-1/ICAM-1 system are not statistically significantly differentfrom zero (P value 0.23) except in one instance (P value =0.06). By comparison, a vast majority of the p values for theTCR/pMHC system are statistically significantly greater thanzero, whereas half of the p values for the C-cadherin systemare statistically significantly less than zero and are markedwith asterisks on the top of corresponding solid bars to indicateP value 0.05.
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